OFFSET
0,3
COMMENTS
The common digits might include leading 0's (such as at n = 5) and they are discarded (in particular, a(0) = 0 indicates that the corresponding zero digit term results in a 0 integer entry).
a(k*10) = 0 for every positive integer k, since (k*10)^((k*10)^(k*10)) and (k*10)^((k*10)^((k*10)^(k*10))) have in common only their rightmost (k*10)^(k*10) digits.
LINKS
Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Joyce Sequence.
Wikipedia, Knuth's up-arrow notation.
FORMULA
a(n) = A002488(n) (mod 10^k), where k is such that n^(n^n) == n^(n^(n^n)) (mod 10^k) and n^(n^n) <> n^(n^(n^n)) (mod 10^(k+1)).
EXAMPLE
For n = 3, 3^(3^3) = 7625597484987 and 3^(3^(3^3)) == 387 (mod 1000) so there are two common final digits a(3) = 87.
CROSSREFS
KEYWORD
sign,base
AUTHOR
Marco Ripà, Jan 27 2024
STATUS
approved